Volume 10, Issue 2

Given a split extension W ⋊ G , where W is an arbitrary Coxeter group and G a group of automorphisms of the Coxeter graph of W , we determine all involutions in W ⋊ G whose centralizers are of finite index. Our result has applications to many problems such as the isomorphism problem for general Coxeter groups. In the course of the proof, some properties of certain special elements and of fixed-point subgroups of graph automorphisms in Coxeter groups which are of independent interest are given.

Following [J. I. Brenner and J. Wiegold. Two-generator groups, I. Michigan Math. J . 22 (1975), 53–64.], we define the exact spread of a group G to be the greatest integer r for which, given any r non-identity elements of G , one can find y ∈ G such that 〈 x i = y 〉 = G for i = 1, 2, …, r . In this paper, we determine the exact spread of M 11 , the smallest sporadic simple group of Mathieu.

We consider direct summands of monomial modules for group algebras over local rings of positive residue characteristic p . If the base ring is a local domain that contains a root of unity of sufficiently large order, the Krull–Schmidt theorem holds for these modules and their representation rings over two such base rings are canonically isomorphic.

We prove that a finite group with nilpotent subgroup H and H -connected transversals is solvable. The proof depends on the classification of finite simple groups.

A finite group G is said to be a 𝒫𝒮𝒯-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G , it is always the case that H is Sylowpermutable in G . A group G is a 𝒯*-group if, for subgroups H and K of G with H normal in K and K normal in G , it is always the case that H is Sylow-permutable in G . In this paper, we show that the classes of finite 𝒫𝒮𝒯-groups and finite 𝒯*-groups coincide. A new characterization of soluble 𝒫𝒮𝒯-groups is also presented.

We introduce a new subgroup embedding property in a finite group called c *-normality. We investigate the influence of c *-normal maximal subgroups on the p -nilpotency, p -supersolvability and supersolvability of finite groups. Our results unify and generalize some earlier results.

For a group H we denote by δ( H ) the number of conjugacy classes of non-cyclic subgroups. Groups all of whose soluble subgroups H satisfy δ( H ) ≤ 2 are classified. It is also shown that the derived length and Fitting length of a soluble group G are bounded by functions of δ( G ).

We prove that every infinite group admits a non-discrete zero-dimensional Hausdorff topology with continuous translations and inversion.

Let Γ be a group which is an extension of ℤ 2 by ℤ, where the action of ℤ on ℤ 2 is given by a structure matrix of the form for some integer k ≥ 2. The group Γ is generated by the set S which consists of the standard basis of ℤ 2 together with a generator of ℤ. We prove that the growth series of Γ with respect to the generating set S is a rational function and we explicitly compute it in terms of k .